3.8 - 4 billion years ago - The Late Heavy Bombardment: a period
during which the Earth, Moon, Venus and Mars were subjected to many
asteroid impacts, after a relatively calm period of several million
years.

In what follows I give the temperature of outer space
in Kelvin at various times. Kelvin means
"degrees Celsius above absolute zero". The
melting point of water is 273 Kelvin; the boiling point is
373 Kelvin.

1.1 billion years from now - The Sun
becomes 10% brighter than today. Runaway greenhouse effect may evaporate
the Earth's oceans. If so, the water in upper atmosphere will
photodissociate and the hydrogen will sail off into outer space.

2 billion years from now - The Andromeda Galaxy approaches our galaxy, the Milky Way:

3.5 billion years from now - The Sun becomes 40% brighter than today.
Conditions on Earth resemble those on Venus today.

3.85 billion years from now - The Andromeda Galaxy makes its first pass by the Milky Way, leading to a burst of star formation:

3.9 billion years from now - Star formation reaches its peak as Andromeda Galaxy makes its first pass by the Milky Way:

4 billion years from now - the Andromeda
Galaxy becomes tidally stretched and the Milky Way is warped after
their first near-collision:

5.1 billion years from now - the cores of the Andromeda
Galaxy and Milky Way will be visible as bright lobes in the sky
during their second pass. There will be less star formation, because
much of the interstellar gas and dust was used up by in previous burst
of star formation.

5.4 billion years from now - The Sun's core runs out of hydrogen,
and it enters its first red giant phase, becoming 1.6 times bigger and
2.2 times brighter than today.

6.5 billion years from now - The Sun becomes a full-fledged red giant,
170 times bigger and 2400 times brighter than today.

6.7 billion years from now - The Sun starts fusing helium and
shrinks back down to 10 times bigger and 40 times brighter than today.

6.8 billion years from now - The Sun runs out of helium and, too
small to start fusing carbon and oxygen, enters a second red phase.
It is 180 times bigger and 3000 times brighter than today.

6.9 billion years from now - The Sun begins to pulsate every
100,000 years, ejecting more and more mass in each pulse, and
finally throwing off all but the hot inner core, becoming a
white dwarf.

7 billion years from now - the cores of the Andromeda Galaxy and Milky Way merge to form a large elliptical galaxy. The aging population of stars is no longer concentrated along a plane, but instead spread out in an ellipsoid. Star formation diminishes.

1013 years from now - The smallest and longest-lived stars capable
of supporting fusion today, red dwarf stars with a mass about 0.08 times that of
the Sun, run out of hydrogen.

1014 years from now - All normal star formation processes cease.
The universe settles down
with a population of stars consisting of about 55% white dwarfs, 45% brown
dwarfs and a small number of neutron stars and black holes. Star formation
continues at a very slow rate due to collisions between brown and/or white
dwarfs.

1017 years from now - All currently existing
white dwarf stars cool to black dwarfs with a temperature of at most 5 Kelvin.

1019 years from now - All galaxies "boil
off", gradually losing their dead stars to intergalactic space.

3 × 1022 years from now - All
binary brown stars spiral in and collide due to gravitational radiation.

1023 years from now - All galactic clusters boil off.
Temperature: 10-13 Kelvin.

From then on: the Universe expands exponentially and cools down
to a temperature of 10-30 Kelvin.
All black holes eventually evaporate, and all
other forms of matter eventually disperse into individual
elementary particles.

Note: the future sounds pretty boring here, but that's because
I've had to omit all the exciting but unpredictable
twists due to future discoveries by various forms of
intelligent life. We can't really tell what
will happen! What you do may have an effect - so be good.

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References for Further Study

Here's the best graphical history of the history I know - a truly
fun book, packed with pictures, charts, and cool facts: